To restrict those solutions further note that \(A\) , in general, is not integrable (its Nijenhuis tensor does not vanish),
and there are more constraints coming from (), and also (REF ). Namely, passing from \(A\)  to
\(A^{\prime }=\frac{d}{dx}A-\frac{1}{2}AJ-\frac{1}{2}J^*A\) , where \(J^k_j=\frac{\partial f^k}{\partial \dot{y}^j}\)  and \(*\) 
is conjugation with respect to \(\phi \)  we get the equation \(\phi (A^{\prime }\xi ,\eta )=\phi (\xi ,A^{\prime }\eta )\) .
One can further iterate this recursive generation of constraints, and this is what is done in [1]}
for \(n=2\) . However, as we will see, for \(n>2\)  already the first iteration is generically sufficient.
